Problem: Simplify the following expression: $q = \dfrac{-54p + 81}{81p - 117}$ You can assume $p \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-54p + 81 = - (2\cdot3\cdot3\cdot3 \cdot p) + (3\cdot3\cdot3\cdot3)$ The denominator can be factored: $81p - 117 = (3\cdot3\cdot3\cdot3 \cdot p) - (3\cdot3\cdot13)$ The greatest common factor of all the terms is $9$ Factoring out $9$ gives us: $q = \dfrac{(9)(-6p + 9)}{(9)(9p - 13)}$ Dividing both the numerator and denominator by $9$ gives: $q = \dfrac{-6p + 9}{9p - 13}$